Recently, a great deal of attention has been focused on the construction of exponential integrators for semi-linear problems. In this paper we describe a matlab package which aims to facilitate the quick deployment and testing of exponential integrators, of Runge-Kutta, multistep and general linear type. A large number of integrators are included in this package along with several well known examples. The so-called ϕ-functions and their evaluation is crucial for stability and speed of exponential integrators, and the approach taken here is through a modification of the scaling and squaring technique; the most common approach used for computing the matrix exponential.

A commutative but not cocommutative graded Hopf algebra H N , based on ordered (planar) rooted trees, is studied. This Hopf algebra is a generalization of the Hopf algebraic structure of unordered rooted trees H C , developed by Butcher in his study of Runge-Kutta methods and later rediscovered by Connes and Moscovici in the context of noncommutative geometry and by Kreimer where it is used to describe renormalization in quantum field theory. It is shown that H N is naturally obtained from a universal object in a category of noncommutative derivations and, in particular, it forms a foundation for the study of numerical integrators based on noncommutative Lie group actions on a manifold. Recursive and nonrecursive definitions of the coproduct and the antipode are derived. The relationship between H N and four other Hopf algebras is discussed. The dual of H N is a Hopf algebra of Grossman and Larson based on ordered rooted trees. The Hopf algebra H C of Butcher, Connes, and Kreimer is identified as a proper Hopf subalgebra of H N using the image of a tree symmetrization operator. The Hopf algebraic structure of the shuffle algebra H Sh is obtained from H N by a quotient construction. The Hopf algebra H P of ordered trees by Foissy differs from H N in the definition of the product (noncommutative Date concatenation for H P and shuffle for H N ) and the definitions of the coproduct and the antipode, however, these are related through the tree symmetrization operator.

The conjunction fallacy, in which individuals report that the conjunction of two events is more rather than less likely to occur than one of the events alone, is a robust phenomenon. Weassessed the possibility that an analysis in terms of functional measurement methodology might be consistent with occurrence of the fallacy. A 3 x 3 design in which we varied the judged likelihood of the two components constituting the conjunction permitted us to assess the possibility that subjects judge the likelihood of conjunctions by averaging the likelihood of their component parts. The results were consistent with this possibility, and this interpretation was supported by analysis of the results in terms of functional measurement methodology.Continued interest in apparent anomalies of human judgment and decision making has not always resulted in definitive accounts of why they occur. The present report addresses one particularly intriguing such anomaly, the conjunction fallacy in probability judgment discussed at length by Kahneman (1982, 1983) and in more recent treatments ofdecision making (e.g., Abelson, Leddo, & Gross, 1987;Birnbaum, Anderson, & Hynan, 1990;Fantino & Stolarz-Fantino, 1991;Gavanski & Roskos-Ewoldson, 1991; Gigerenzer, in press;Massaro, 1994; Stolarz-Fantino & Fantino, 1990;Stolarz-Fantino, Fantino, & Kulik, 1996;Wells, 1985;Yates & Carlson, 1986). Subjects evincing the conjunction fallacy report that the conjunction oftwo events is more rather than less likely to occur than one of the events alone. This phenomenon is illustrated by an example used in the present research (see Appendix); the example is patterned after those in Tversky and Kahneman (1982).Logically, Ralph is at least as likely to play in a heavy metal band for a hobby as he is to play in a heavy metal band for a hobby and happen to be a building inspector. Yet "Ralph plays in a heavy metal band" is judged less likely by most subjects than the conjunction. The robustness ofthis effect is well documented in Tversky and Kahneman (1983) and by our own attempts to eliminate it. For example, Stolarz-Fantino and Fantino (1990) reasoned that if instructions provided discriminative stimuli This research was conducted in accordance with human subjects guidelines at this institution. Informed consent was obtained from the subjects.

Observations in rocky intertidal areas demonstrate that breaking waves 'throw' rocks and cobbles and that these missiles can damage and kill organisms. Targets in the intertidal were dented by impacts from wave-borne rocks. New dents/day in these targets was positively correlated with the daily maximum significant wave height and with new patches/day in aggregations of the barnacle Chthamalus fissus. Impact frequency was highest in the upper intertidal and varied dramatically between microhabitats on individual boulders (edges, tops and faces). These patterns were reflected in the microhabitat abundances of 'old' and 'young' barnacles. Comparisons were made of the survivorship and the frequency of shell damage in two populations of the limpet Lottia gigantea living in habitats which differed primarily in the number of moveable rocks (i.e. potential projectiles). The mortality rate and frequency of shell damage were significantly higher in the projectilerich habitat. In addition only in this habitat did the frequency of shell damage covary significantly with seasonal periods of high surf. Investigation of the response of limpet shells to impacts suggests that shell strength varies between species and increases with shell size. Species-specific patterns of non-fatal shell breakage may have evolved to absorb the energy of impacts. In two of the intertidal habitats studied, wave-borne rock damage was chronic and, at least in part, may have governed the faunal makeup of the community by contributing to the physical 'boundaries" of the environment within which the inhabitants must survive.

Abstract.We show that while Runge-Kutta methods cannot preserve polynomial invariants in general, they can preserve polynomials that are the energy invariant of canonical Hamiltonian systems.Mathematics Subject Classification. 65P10, 65L06. All Runge-Kutta (RK) methods preserve arbitrary linear invariants [12], and some (the symplectic) RK methods preserve arbitrary quadratic invariants [4]. However, no RK method can preserve arbitrary polynomial invariants of degree 3 or higher of arbitrary vector fields [1]; the linear systemẋ = x,ẏ = y,ż = −2z, with invariant xyz, provides an example. (Preservation would require R(h) 2 R(−2h) ≡ 1, where R is the stability function of the method 4 ; but this requires R(h) = e h , which is impossible [6].) This result does not rule out, however, the existence of RK methods that preserve particular (as opposed to arbitrary) invariants. Since the invariant does not appear in the RK method, this will require some special relationship between the invariant and the vector field. Such a relationship does exist in the case of the energy invariant of canonical Hamiltonian systems: see [3,5] on energy-preserving B-series. In this article we show that for any polynomial Hamiltonian function, there exists an RK method of any order that preserves it.The key is the average vector field (AVF) method first written down in [9] and identified as energy-preserving and as a B-series method in [10]: for the differential equatioṅthe AVF method is the map x → x defined by